The topic of function spaces with a variable exponent is a lively area of research at present, mainly due to its wide applications in the modeling of electrorheological fluids, in the study of image processing, and differential equations with non-standard growth. These aforementioned applications are mainly in Lebesgue and Sobolev spaces with variable exponent. Different aspects concerning these spaces with variable exponents can be found in [1]} and [2]} and in their bibliography.
